Problem 49 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ 4(2 n+1)=3(6 n+3)+1 $$

Step-by-Step Solution

Answer

The solution is \( n = -\frac{3}{5} \).

Step 1: Distribute the Multiplication

First, distribute the numbers outside the parentheses to each term inside the parentheses.On the left side: \[ 4(2n + 1) = 8n + 4 \] On the right side: \[ 3(6n + 3) = 18n + 9 \] So, the equation transforms to: \[ 8n + 4 = 18n + 9 + 1 \]

Step 2: Simplify Both Sides

Now simplify each side of the equation.On the right side, add 9 and 1: \[ 18n + 9 + 1 = 18n + 10 \] Thus, the equation becomes: \[ 8n + 4 = 18n + 10 \]

Step 3: Isolate the Variable Term

To isolate the variable term, subtract \(8n\) from both sides of the equation.\[ 8n + 4 - 8n = 18n + 10 - 8n \] This simplifies to: \[ 4 = 10n + 10 \]

Step 4: Eliminate the Constant Term

Subtract 10 from both sides to eliminate the constant on the right side:\[ 4 - 10 = 10n + 10 - 10 \] Simplifying gives: \[ -6 = 10n \]

Step 5: Solve for the Variable

Finally, divide both sides by 10 to solve for \(n\).\[ n = \frac{-6}{10} \]This reduces to: \[ n = -\frac{3}{5} \]

Step 6: Conclusion

The value of \(n\) that satisfies the given equation is \(-\frac{3}{5}\).

Key Concepts

Understanding the Distributive Property

The distributive property is a key concept in algebra essential for simplifying expressions. It allows you to remove parentheses in expressions involving multiplication over addition or subtraction. Here's how it works: For any numbers or expressions \[ a(b + c) = ab + ac \]This means that you multiply the number outside the parentheses by each term inside. In the given equation, we have:

• The left side: \[ 4(2n + 1) \] becomes \[ 8n + 4 \]
• The right side: \[ 3(6n + 3) \] becomes \[ 18n + 9 \]

Using the distributive property helps us simplify these terms and eliminate the parentheses. This is crucial since it gives us a clearer view of the equation so we can solve it step by step.

Applying Steps to Solve Equations

Solving equations is about finding the unknown value that makes the equation true. In our case, this unknown value is the variable \( n \). Let's break down our approach:- **Simplify First:** After applying the distributive property, simplify any constants on each side. For example, in the equation \( 8n + 4 = 18n + 9 + 1 \), you add 9 and 1 to simplify to \( 18n + 10 \).- **Isolate Variable:** Start moving terms to isolate the variable on one side. This usually means getting all terms with the variable on one side and constants on the other.With the equation \( 8n + 4 = 18n + 10 \), isolating \( n \) involves moving terms around until \( n \) is by itself on one side.

Mastering Variable Isolation

Variable isolation is a technique used to solve equations by getting the variable alone on one side of the equation. Here's how you do it:- **Rearrange Terms:** You need to manipulate the equation to group all terms with \( n \) on one side and numbers on the other. In our example, subtract \( 8n \) from both sides: \[ 8n + 4 - 8n = 18n + 10 - 8n \] Which simplifies to \[ 4 = 10n + 10 \].- **Remove Constants:** Next, subtract other numbers hindering \( n \) from being isolated. Subtract 10 from both sides: \[ 4 - 10 = 10n \] Which gives \[ -6 = 10n \].- **Solve for \( n \):** Finally, divide both sides by 10 to solve for \( n \): \[ n = \frac{-6}{10} = -\frac{3}{5} \].By isolating the variable, you can clearly find its value. Practice makes perfect, and these steps help simplify the equation-solving process.